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Abstract: In this article, we consider planar graphs in which each vertex is not incident to some cycles of given lengths, but all vertices can have different restrictions. This generalizes the approach based on forbidden cycles which corresponds to the case where all vertices have the same restrictions on the incident cycles. We prove that a planar graph is 3-choosable if it is satisfied one of the following conditions: [(1)] has no cycles of length 4 or 9 and no 6-cycle is adjacent to a 3-cycle. Moreover, for each vertex , there exists an integer such that is not incident to cycles of length . [(2)] has no cycles of length 4, 7, or 9, and for each vertex , there exists an integer such that is not incident to cycles of length . This result generalizes several previously published results (Zhang and Wu, 2005 , Chen et al., 2008 , Shen and Wang, 2007 , Zhang and Wu, 2004 , Shen et al., 2011 ). [Copyright &y& Elsevier]

Abstract: A 3-simplex is a collection of four sets with empty intersection such that any three of them have nonempty intersection. We show that the maximum size of a set system on elements without a -simplex is for all , with equality only achieved by the family of sets containing a given element or of size at most . This extends a result of Keevash and Mubayi, who showed the conclusion for sufficiently large. [Copyright &y& Elsevier]

Abstract: Consider the unit circle with distance function measured along the circle. We show that for every selection of points there exists such that . We also discuss a game theoretic interpretation of this result. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we study graph embeddings in pseudosurfaces formed by three spheres sharing at most two points each pair, and in such a way that all vertices in the graph are in the same face. Our results and examples show that the behaviour of outer embeddings in these pseudosurfaces is rather different from embeddings on the pseudosurfaces considered in the literature so far. [ABSTRACT FROM AUTHOR]

Abstract: Three-level secret sharing schemes arising from the vector space construction over a finite field are presented. Compared to the previously known schemes, they allow a larger number of participants with respect to the size of . The key tool is the twisted cubic of . [Copyright &y& Elsevier]

Abstract: We consider point–line geometries having three points on every line, having three lines through every point (bislim geometries), and containing triangles. We classify such geometries under the hypothesis of the existence of a collineation group acting transitively on the point set. [Copyright &y& Elsevier]

Abstract: We give a construction of -regular graphs of girth using only geometrical and combinatorial properties that appear in any -cage, a minimal -regular graph of girth . In this construction, and are odd integers, in particular when is a power of 2 and we use the structure of generalized polygons. With this construction we obtain upper bounds for the -cages. Some of these graphs have the smallest number of vertices known so far among the regular graphs with girth . [Copyright &y& Elsevier]